ieee journal on selected areas in communications, vol. 24 … · 1582 ieee journal on selected...

IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 8, AUGUST 2006 1581

Optimizing Power Efficiency of OFDM UsingQuantized Channel State Information

Antonio G. Marqués, Student Member, IEEE, Fadel F. Digham, Member, IEEE, andGeorgios B. Giannakis, Fellow, IEEE

Abstract—Emerging applications involving low-cost wire-less sensor networks motivate well optimization of orthogonalfrequency-division multiplexing (OFDM) in the power-limitedregime. To this end, the present paper develops loading algorithmsto minimize transmit-power under rate and error probabilityconstraints, using three types of channel state information at thetransmitter (CSIT): deterministic (per channel realization) forslow fading links, statistical (channel mean) for fast fading links,and quantized (Q), whereby a limited number of bits are fed backfrom the transmitter to the receiver. Along with optimal bit andpower loading schemes, quantizer designs and reduced complexityalternatives with low feedback overhead are developed to obtaina suite of Q-CSIT-based OFDM transceivers with desirable com-plexity versus power-consumption tradeoffs. Numerical examplescorroborate the analytical claims and reveal that significant powersavings result even with a few bits of Q-CSIT.

Index Terms—Frequency-selective channels, interpolation tech-niques, low-power systems, orthogonal frequency-division multi-plexing (OFDM), optimization methods, quantization techniques.

I. INTRODUCTION

ORTHOGONAL FREQUENCY-DIVISION MULTI-PLEXING (OFDM) has been the “workhorse modula-

tion” for bandwidth-limited wireline and wireless transmissionsover frequency-selective multipath channels. Testament to itswell-documented merits is provided by the fact that OFDMhas been adopted by digital subscriber line (DSL) modems,digital audio and video broadcasting (DAB/DVB) standards,and wireless local area networks, to name a few [12], [19].Spectral efficiency and error resilience in these applications arewell known to improve with the knowledge of channel stateinformation at the transmitter (CSIT). For this reason, OFDMtransmissions over wireline or slowly fading wireless links havetraditionally relied on deterministic (per channel realization)CSIT to adaptively load power and bits so as to maximizerate (capacity) for a prescribed transmit-power. However,errors in estimating the channel at the receiver, feedback delay,and the asymmetry between forward and reverse links renderacquisition of deterministically perfect CSIT pragmatically

Manuscript received September 19, 2005; revised April 15, 2006. The workof A. G. Marqués was supported in part by the Spanish Government under GrantTEC2005-06766-C03-01/TCM. This work was supported in part by the USDoDARO under Grant W911NF-05-1-0283.

A. G. Marqués is with the Department of Signal Theory and Communications,Rey Juan Carlos University, Cmno. del molino s/n, Fuenlabrada, Madrid 28943,Spain (e-mail: [email protected]).

F. F. Digham and G. B. Giannakis are with the Department of Electrical andComputer Engineering, University of Minnesota, Minneapolis, MN 55455 USA(e-mail: [email protected]; [email protected]).

Digital Object Identifier 10.1109/JSAC.2006.879397

impossible in most wireless scenarios. This has motivatedOFDM loading schemes based on statistical (S) CSIT or quan-tized (Q) CSIT, whereby a limited number of bits are fed backfrom the receiver to the transmitter; see, e.g., [6], [8], [14] and[20], and references therein. But these works too deal with thebandwidth-limited setup, where the objective is to maximizerate or minimize bit-error rate (BER).

Interestingly, except for [17] where deterministic (D) CSITis used to minimize power consumption, analogous efforts havenot been devoted to optimizing OFDM in the power-limitedregime. This is the theme of the present paper which aims atloading algorithms to minimize transmit-power under rate anderror probability constraints using Q-CSIT. This goal is timelyfor both commercial, as well as tactical communication systemsdesigned to extend battery lifetime, and especially for wirelessnetworks of sensors equipped with nonrechargeable batteries.Our focus on Q-CSIT is well justified since the resultant trans-ceivers are universally applicable to frequency-selective wire-less channels, they incur controllable amount of feedback over-head and implementation complexity, they turn out to be morepower-efficient than S-CSIT-based ones, and for a sufficientnumber of feedback bits they even approach the power savingsachieved by the benchmark D-CSIT designs.

The rest of this paper is organized as follows. After intro-ducing preliminaries on the setup we deal with (Section II),we derive optimal power and bit loading schemes for OFDMbased on D-CSIT in order to provide the fundamental limit inpower efficiency (Section III). Subsequently, we develop themost widely applicable OFDM transceivers based on Q-CSITwhich entail two design phases: the offline phase specifying theQ-CSIT format by selecting channel quantization thresholdsbased on channel mean statistics, and the online one derivingthe bit and power loading schemes per OFDM subchannel(Section IV). A suite of practical derivatives are developednext trading-off power savings for low complexity (Section V)and reduced feedback overhead (Section VI). To complete thegamut of pertinent options, power-efficient OFDM schemesbased on S-CSIT are also outlined (Section VII). Finally,numerical results and comparisons of the various alternativesare presented (Section VIII), and concluding remarks wrap upthis paper (Section IX).

II. PRELIMINARIES AND PROBLEM STATEMENT

With reference to Fig. 1, we consider wireless OFDM trans-missions over subcarriers (subchannels) through frequency-selective fading channels with discrete-time baseband equiva-lent impulse response taps , wheredenotes the channel order, the maximum delay spread,

0733-8716/$20.00 © 2006 IEEE

1582 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS, VOL. 24, NO. 8, AUGUST 2006

Fig. 1. Transmission system block diagram.

the transmit-bandwidth, and stands for the floor opera-tion. A serial stream of data bits is demultiplexed to formparallel streams each of which (indexed by )is multiplied by an appropriate constant to load power , andmodulated to obtain symbols drawn from a constel-lation of size . The -point inverse fast Fourier transform(I-FFT) applied to each snapshot of the symbol streams isfollowed by insertion of a size- cyclic prefix (CP) to yield ablock of symbols (also known as OFDM symbol), whichare subsequently multiplexed and digital to analog (D/A) con-verted for transmission. These operations along with the corre-sponding FFT and CP removal at the receiver convert the mul-tipath fading frequency-selective channel to a set of parallelflat-fading subchannels each with fading coefficient given by theFFT (see, e.g., [19]): ,where is typically chosen so that .

Channel estimation at the receiver relies on periodically in-serted training symbols (also known as pilots). With the channelacquired, the receiver has available a noise-normalized channelgain vector , where denotes transpo-sition and is the instantaneous noise-normal-ized channel gain of the th subchannel on which the zero-meanadditive white Gaussian noise (AWGN) has variance . Eachdeterministic realization of the random gain constitutes theD-CSI, while its statistical average is what wehenceforth refer to as S-CSI of the th subchannel ( denotesthe expectation operator). Either one must be available to thetransmitter in the D-CSIT or S-CSIT-based OFDM designs, but

will also be used in specifying the form of Q-CSIT; i.e., thechannel quantizer employed by the Q-CSIT-based OFDM trans-ceivers. The quantizer design entails selection of quantizationthresholds , which in turn define quantization regionseach subchannel realization falls into. The random vector ofbits representing indices of the regionsof all subchannels is what we term Q-CSI, and is what must befed back from the receiver to the transmitter in Q-CSIT-basedOFDM schemes.

Given either (D-CSIT), or its mean (S-CSIT), or the quan-tized form of the channel gains specified by the bit vector(Q-CSIT), our ultimate goal is to choose power and bit vectors

and so as to mini-mize the power , or its average , for prescribedvalues of the rate and the bit error rate per OFDM

symbol . In addition, we wish to choosejudiciously the channel quantizer in Q-CSIT-based designs, anddevelop bit and power loading algorithms which are flexible toachieve desirable tradeoffs among implementation complexity,the required feedback overhead, and power efficiency. We willpursue these objectives under the following assumptions.

a1) Symbols are drawn from quadrature amplitude mod-ulation (QAM) constellations of size .

a2) Subchannels remain invariant over at least twoconsecutive OFDM symbols, they are allowed tobe correlated, and each is complex Gaussian dis-tributed; i.e., each subchannel gain adheres toan exponential probability density function (pdf)

.a3) Feedback channel is error-free and incurs negligible

delay.Assumptions a1–a3 are common to existing designs [6], [8],

[12], [14], and are typically satisfied in practice. In principle, ourresults apply to any channel pdf, but a2 simplifies the resultantdesigns. Channel invariance in a2 allows for feedback delays; itmay be stringent for D-CSIT-based designs, but is very reason-able in the Q-CSIT case since each subchannel can vary fromone OFDM symbol to the next so long as the quantization re-gion it falls into remains invariant. Finally, error-free feedbackis easily guaranteed with sufficiently strong error control codesespecially since rate in the reverse link is low.

With denoting rate (number of bits per OFDM block), theinstantaneous BER is

(1)

where and forthe QAM constellations in a1, we have ,

, with if , or, if ;and the approximation of the corresponding to the thsubchannel is very accurate [5], [9]. Notice that for a given

, we can express as a function of and the modula-tion related constants and . This expression as well as (1)will come handy in the next section, where we optimize bit andpower loading based on D-CSIT.

MARQUÉS et al.: OPTIMIZING POWER EFFICIENCY OF OFDM USING QUANTIZED CHANNEL STATE INFORMATION 1583

III. OPTIMIZATION BASED ON D-CSIT

In this section, we rely on the D-CSIT gain vector to op-timize power efficiency of OFDM for slow fading wireless orwireline channels. Since represents the full CSI possibly avail-able, the resultant power and bit allocation also serve as bench-marks for the designs based on partial (namely quantized or sta-tistical) CSI. With denoting the finite set of possible valuesof , our objective can be formulated as follows:

(2)

Since entries in take on continuous values while those inare integer-valued, we deal with a mixed-integer programming(MIP) problem, which can be solved in two steps. First, we willsolve (2) for a fixed (i.e., with C2 removed and the optimum

expressed in terms of ), and then we will select the overalloptimum pair by exhaustively testing over all finite re-alizations of .1 Substituting (1) into (2), the first step is

(3)

Invoking the necessary Karush–Kuhn–Tucker (KKT) condi-tions for optimality [10, Sec. 14.6], the solution can befound to be

(4)

where , and is the so-called Lagrangemultiplier which can be obtained after plugging (4) into C1.Notice that apart from , only depends on , i.e.,

. Because both and C1 are convex func-tions of , the solution in (4) is globally optimum. Reminiscentof the water-filling principle, (4) also shows that depending onsubchannel gains, it is possible for certain subchannels to re-ceive zero-power. Let be the set of indices corresponding tothe active subchannels which are allocated nonzeropower. Upon substituting (4) into C1, we obtain

(5)

1As will be shown shortly in Algorithm 1, the search space over m can berendered very confined.

We next incorporate the rate constraint C2 in the second stepof our optimization process, and reduce the aforementioned ex-haustive search over following the algorithm described next.

Algorithm 1

D1) Sort the subchannels in descending order according totheir values.

D2) Create a table of all possible vectors that satisfy theconstraint C2 ordered according to subchannel gains;i.e., if , then .

D3) For each of the table, evaluate the correspondingwith entries as in (4), and select the pair thatleads to the minimum .

Sorting in D1 confines the search space over considerably;see also [17] for a proof. The intuition behind D1 is that thehigher is, the higher can be loaded to minimize .

Similar to the classical water-filling-based loading schemewhere rate is maximized subject to a power constraint [3], inour loading scheme for power optimization subject to rate andBER constraints, generally increases for larger which inturn causes to increase. But for subchannels loaded with thesame , power decreases as increases.

To gain further insight about our solution of (3), let us con-sider the related problem of minimizing under a constrainton capacity. In this case, the KKT conditions readily yield theoptimum power allocation as

(6)

where , and is the prescribedcapacity. It is interesting to notice that absence of the BER con-straint here, causes to always increase as increases.

Remark 1: Alternative MIP techniques accounting jointlyfor C1 and C2 can in principle be used to solve the optimiza-tion problem in (2). However, joint consideration of C1 andC2 increases the problem dimensionality and the optimizationburden. For instance, applying the branch-and-bound algorithmwith Lagrangian relaxation to solve (2) requires solving a set ofnonconvex subproblems [1].

Remark 2: In the optimal loading schemes of this sectionas well as those of the ensuing sections, we did not imposepeak-to-average-power-ratio (PAPR) constraints. However,available digital predistortion schemes (such as the selectedmapping algorithm in [18]) can be applied to each blockof symbols in order to meet prespecified PAPRconstraints.

IV. OPTIMIZATION BASED ON Q-CSIT

Instead of optimizing power efficiency of OFDM based onthe realization of the subchannel gains, here we rely on the bitvector comprising codewords which represent the quanti-zation region indices each subchannel gain falls into. In thiscontext, we will assume that:


a4) Codeword has length ; hence, is anvector with 0,1 entries.

These bits per subchannel will index regions, that the space is divided using quantization

thresholds for each subchannel. Notice that for brevity, we use (rather than )

to index regions and thresholds; however, since the latter includealso the subscript , we should keep in mind that regions andthresholds are allowed to be different from one subchannel toanother. Further, since quantization here pertains to gains, weset and .

Our Q-CSIT-based optimization will proceed in two phases:1) specification of the optimum thresholds , which canbe performed offline and 2) derivation of the optimum powerand bit loading vectors to be used online based on theQ-CSI codeword vector obtained in the quantizer design phase1).

A. Offline Phase: Quantizer Design

We wish to derive quantization thresholds for the randomvariable which has a known pdf as per a2. Those shouldclearly depend on the underlying distortion metric which,in par with our objective on power efficiency, will be takento be the average power . To evaluate , recallthat for a prescribed BER the power will depend on thequantized , which takes possible values; correspondingly,

is a discrete random variable taking valueswith probability

,where the last equality comes from the exponential pdf in a2. Wecan thus readily deduce that ,which shows how the average power depends on the thresholdsand the channel pdf.

Arguing likewise, since is a continuous random vari-able and is a discrete one, we can also express theaverage BER per subchannel as:

. Notice thatfor the region corresponding to , we have

. For the quantization thresholds to be suitable across channelrealizations, we will use the average power as objective func-tion and include the average BER in our constraints. Thus,our channel quantizer design for selecting thresholds can beformulated as follows:

(7)

As in our D-CSIT-based optimization, we will first solve (7)without accounting for C2. Upon substituting the expressionsfor and into (7) and defining ,

the objective function and constraint C1 for this relaxedproblem can be written, respectively, as

(8)

(9)

Lack of convexity that we demonstrate in Appendix A impliesthat only locally optimal solutions can be ensured in our quan-tizer design. Nonetheless, the Lagrangian for minimizing (8)subject to (9) is given by

(10)

where is found after fulfilling (9). Upon defining, the necessary

conditions and , respectively,yield (for and )

(11)

(12)

Equations (11) and (12) can be compactly written asnonlinear functions of the powers and thresholds as

and ,respectively. We solve them to obtain using the followingoffline algorithm.

Algorithm 2

Q1) Sort the subchannels in decreasing order based onand generate a table of possible candidates satisfyingthe rate constraint as in steps D1 and D2 of Algorithm1. Then, execute the following loop for each entry of .

Q2) For values:• For

• For , where is found so thate.g., ;Given a value, solve (11) numerically(e.g., using the bisection method [15]) andobtain a candidate root (recall ).


With given, solve (12) to determine .Similarly, keep alternating between (11) and (12)to eventually obtain and for all

. As these power and thresholdvalues per subchannel are found without using (12)with , this equation can be used to testthe feasibility of the candidate solution obtained.

• End loop.• End loop.• At this point, we have a list of candidate solutions

. It remains to filter these solutions and acceptonly those achieving the average BER constraint in(9).End loop.

Q3) Among all feasible ’s, choose the one leading to theminimum in (8).

In step Q1, each choice of satisfies: 1) and2) if . This means that Algorithm 2 yieldsthe thresholds only for the subchannels that are “best on theaverage”. Since on an OFDM symbol-by-symbol basis, we mayneed thresholds for all subchannels, numerical simulations havesuggested a rule of thumb whereby for each , the ratioremains invariant for all . This rule will lead naturally to oneof the schemes we introduce in the next section.

In step Q2, given an initial value, different algo-rithms can be used to update [1, Ch. 4]. For example,using the method of multipliers entails

,where denotes the iteration index and is the penalty param-eter that can be updated using different ad hoc methods.

Remark 3: In addition to thresholds, Algorithm 2 yields as aby-product loadings minimizing average power subjectto average BER based on statistical CSI (here, the average gains

). This pair can be used in simpler but suboptimumloading algorithms we will explore in Section V-C. But first, wewill see next how instead of , the optimum online algorithmbased on the Q-CSIT codeword can yield improvedloadings given the quantization thresholds.

B. Online Phase: Power and Bit Loading

With the quantizer parameters (thresholds and thus regions)fixed, each gain realization is estimated and quantized at thereceiver to obtain the codeword . Feeding back this form ofinstantaneous Q-CSIT, the transmitter will find online the pairof loadings which minimize power subject to rate, and(what we could call conditioned on ) average BER per sub-channel. With , the latter can be expressedas

(13)

With this conditional average BER, our optimization problemcan be formulated as follows:

(14)

Comparing (14) with (7), we infer that apart from the fact thatthe quantizer design and the regions are given, the structure ofboth problems is similar. Since the scalar in the denominatorof (13) will just alter the multiplier, a simplified version ofAlgorithm 2 which does not search over thresholds and does notaccount for (11) can be applied to solve (14) as well. Further-more, because here we wish to solve for the pair as in Al-gorithm 1 (rather than the thresholds we sought in Algorithm 2),we expect convexity to ensure global optimality. Indeed, the suf-ficient conditions for optimality require , and

. The first condition can be readily verified,while the latter yields

(15)

where the last inequality follows since and theintegrand is non-negative.

V. REDUCED COMPLEXITY OPTIMIZATION BASED ON Q-CSIT

Per iteration of the quantizer design in Algorithm 2 (corre-sponding to each feasible ), we look for thresholds

, and power levels per subchannel. Repeatedfor all feasible , this requires searching over un-knowns. Likewise, the online search space entailsunknowns to solve for in all subchannels for each . As

and increase, this motivates well the reduced complexity,albeit suboptimal, schemes we develop in this section.

A. Reduced Complexity Optimization and Loading

The key to eliminating half of the unknowns in both phases ofthe optimization process is to express power levels in termsof the thresholds . To do so, our approach here is to capitalizeon the optimum [c.f., (4)] obtained for the D-CSIT caseand examine four reduced complexity (RC) alternatives.RC1) Estimate the quantized channel gain as

, and use (4) to obtain .


RC2) Utilize directly the pdf in a2 to estimate channel gainsvia

(16)

and then employ (4) again to obtain power levels as.

RC3) Bypass estimation of the quantized , by using directlythe -dependent power quantizer (see also [4])

(17)

RC4) Rely as in RC2 on the channel pdf, but use instead themore precise power quantizer

, which can be evaluated using (4) andnumerical integration.

Among these four options, RC1 (middle point of the gain quan-tization region) is the simplest, but as simulations confirm it alsoleads to the least power efficient Algorithm 2. On the other hand,RC4 (expected power) is slightly better than the RC2 and RC3but is rather complex as it does not lead to a closed-form expres-sion of in terms of . Eventually, RC3 (power middlepoint) is the most promising when considering both complexityand power efficiency. For this reason, we henceforth use (17)that can be rewritten using (4) as

(18)

where is a constant (that we will term power quantizationparameter), which replaces , and as we will see later has tobe determined jointly with the Lagrange multiplier corre-sponding to the Q-CSIT-based algorithm. Based on (18), wewill simplify the offline phase (by optimizing only over ), aswell as the online phase (by reducing optimization to a one-di-mensional search over ).

B. Reduced Complexity Offline Phase

We develop in this subsection two suboptimum yet compu-tationally efficient schemes that will reduce complexity in thequantizer design phase.

1) Common Thresholds (CT) for All Subchannels: Here, wedrop the threshold subscript and use for all , whichleads to an -fold reduction in the threshold search space (only

unknowns are involved in each iteration of Algorithm 2).Straightforward application of the KKT conditions with carried

out simplifications yields the optimum CT as the solutionof the following equations (for ):

(19)

where is the Lagrange multiplier andif or and

, and zero otherwise.Iterations then proceed as in Algorithm 2 with the exception

that we need two loops to search for the Lagrange multiplierand the power quantization parameter (involved in

and ). Upon solving for the thresholds and , can beobtained using (18). Even though this CT-QCSIT is simple, itmay not effectively exploit the statistics of the subchannel gains.In the next subsection, we derive instead a simpler yet moreefficient scheme to solve for the thresholds.

2) Subcarrier Irrespective Thresholds (SIT): Here, we relyon the rule of thumb we mentioned before Remark 3, whichasserts that for each , for some constant ir-respective of . Notice that unlike CT, this SIT-based designaccounts for the individual subchannel means, and dependingon the distortion metric adopted, we can devise two simplifiedquantizers: one enforcing equiprobable (EP) quantization re-gions, and another minimizing the mean-square error in quan-tizing either the gains (what we naturally term Lloyd- quan-tizer), or, the powers (what we call Lloyd- quantizer).

a) EP quantization: In this case, we determine ’s sothat . Underthe pdf in a2, this yields the following closed-form solution:

(20)

which interestingly adheres to the aforementioned rule that, and asserts a means of optimality to the EP

quantizer. Notice that the closed-form expression (20) reduces


TABLE INUMBER OF ITERATIONS REQUIRED FOR THE DIFFERENT METHODS PROPOSED

complexity also in the online phase since the average BERconstraint in (14) becomes

(21)

b) Lloyd scalar quantization: In this case, we adoptthe scalar Lloyd algorithm [7], [13], which for a randomvariable with pdf quantized to (using levels)is known to minimize . The quantized andthresholds , are obtained by iteratively solving in an al-ternating fashion the equations: and

. We can apply Lloyd’salgorithm to our problem with either or (whatwe referred to as Lloyd- or Lloyd- quantizers).

Remark 4: The SIT schemes designed during the offlinephase are employed by the optimum online algorithm that uses(18) to obtain . Although SIT treats subchannels sep-arately, the online algorithm accounts for subchannels jointlythrough the conditional average BER constraint.

C. Reduced Complexity Online Phase

As we mentioned in Section V-A, (18) can be used in the on-line phase to simplify the search required to satisfy C1 in (14)for each feasible . In Algorithm 2, for each value of , weperform searches using (12) to determine the power alloca-tion and second check C1 with the current Lagrange multiplier,repeating this process until C1 is satisfied. However, using (18),we only need one single search over to satisfy C1, thus re-ducing complexity drastically.

Furthermore, as we mentioned in Remark 3, Algorithm 2yields as a by-product a pair of suboptimum loadings inthe offline phase. These can be used to reduce complexity alsoduring the online phase by having the transmitter operate in anyof the following modes (sorted in decreasing power efficiencybut also in decreasing complexity order).

TxA) Select optimally the set of active subcarriers , bitloading , and the value of the Lagrange multiplier,

, to attain the global optimum .

TxB) Select optimally and while using a fixed (noconstellation adaptation).

TxC) Select optimally , but use fixed values for and .

TxD) Select optimally , but use always the same and.

It is then up to the designer to select among TxA–TxD de-pending on application specific constraints.

D. Complexity Comparison

Let denote the number of possible candidates whichsatisfy the rate constraint, the number of candidate valuesfor the first threshold in each region, the number of iterationsneeded to solve for the Lagrange multiplier associated with theBER constraint, the number of iterations needed to deter-mine each power level using any root-finding method (giventhe values of other parameters), the number of iterationsrequired to obtain one threshold solution and the numberof iterations needed until the Lloyd algorithm converges [c.f.Section V-B2b]. Table I summarizes the computational com-plexity for the various design methods contrasting online andoffline phases and illuminating the potential reduction whenany of the RC alternatives is employed. As shown, using CTwith an RCx alternative (specifically with RC3 as analyzed inSection V-B1) reduces complexity considerably. For the SITcase, the thresholds can be obtained directly or in conjunctionwith . Correspondingly, the notation introduced inTable I reduces to when we are only interested in the quan-tizer design and reduces to when are also evaluated onaverage to be used in the online phase by Tx-B, Tx-C, or Tx-D.More importantly, Table I reveals that the online complexity canbe as simple as selecting the active subcarriers or just solving forone unknown, namely, .

So far, we have been using bits of feedback per subchannelfor a total of feedback bits, which may be prohibitive asincreases and the channel coherence time decreases. To handlesuch cases, we will exploit the statistical dependence amongsubchannels (especially when ) and introduce in theensuing section an efficient means of reducing the feedbackoverhead.

VI. Q-CSIT WITH REDUCED FEEDBACK OVERHEAD

It can be readily verified that points on the discreteFourier transform (DFT) grid suffice to identify thenonzero taps. Based on this fact, we prove in Appendix Bthe following.

Proposition 1: Only bits suffice to quantize thechannel gain vector .


Fig. 2. Interdependence computation modules.

For , this reduces the required number of feedbackbits considerably: from to . Becausesamples of on the DFT grid suffice to fully identify the en-tire gain profile, we let denote the set of the indices of the

uniformly sampled subchannels and its complement.Given , we first form the estimates for , and theninterpolate to obtain the remaining ’s for . Notice thatany of the offline schemes described in the preceding sectioncan be used first to solve for the thresholds in all subchannels.Then, the two aforementioned estimation/interpolation tasks areexecuted at the transmitter during the online phase. Afterwards,any of the online algorithms discussed in the previous sectioncan be implemented. Fig. 2 depicts the different computationmodules along with their interactions. The first task of the es-timation/interpolation module, namely subchannel estimation,can be accomplished by using any of the four RC options dis-cussed in the previous section. In RC3 and RC4, we estimatepower which is then used to estimate for . Taking RC3as an example and having available the thresholds as well as theoffline (while evaluating thresholds) values of the parametersand , can be found via (18) by solving

(22)

These different methods of estimation will be compared inSection VIII.

Given for , our next task is to interpolate and esti-mate for . Various interpolators are possible but ournumerical results suggest that the sinc interpolator leads to themost power efficient scheme. Upon defining

, the sinc interpolator in ourcase yields

(23)

Up to this point, we explored D-CSIT and Q-CSIT-baseddesigns. To complement our study, it remains to investigateschemes relying on statistical CSIT (S-CSIT), which offersthe least complex option and also incurs the lowest feedbackoverhead.

VII. OPTIMIZATION BASED ON S-CSIT

Based on the channel pdf (in our case the mean ), our ob-jective in this section is to determine the power minimizing afixed pair of loadings . We will consider two possibili-ties: in the first one we will optimize over and , while in thesecond we will only optimize over at uniform power loading.

A. Optimum Bit and Power Loading

Our optimization problem here is formulated as follows:

(24)

Averaging in C1 needs to be carried out over all possible valuesof

. Removing temporarily C2, as before, theKKT conditions on the relaxed problem yield

(25)

where . Thesufficient conditions for optimality, and

, are easily shown to hold true, thusensuring global optimality. As discussed earlier, (25) mustbe calculated for all sorted candidates that meet theconstraint, selecting as final solution the one which requiresless power.

B. Optimum Bit and Uniform Power Loading

We finally investigate the possibility of having uniform powerloading over the active subchannels. Let denotethe uniform power value, the indicator function, and define

. In this case,the search space in (24) is confined over and .Since no analytical solution is available, we optimize as beforeover all feasible sorted satisfying C2, and then numericallyfind that satisfies C1. Finally, we choose the optimumpair that leads to the minimum . As is constant acrosssubcarriers, a simpler algorithm using a greedy approach canbe implemented along the lines of [2, Ch. 16] to allocate bits,followed by a numerical search to find the value of .

VIII. NUMERICAL EXAMPLES

To numerically test our power-efficient designs, we con-sidered an adaptive OFDM system with subcarriersand feedback bits per subcarrier (when Q-CSIT isutilized) operating over fading channels with average gains

allowed to vary over a 10 dB range, and prescribed toachieve 10 at rate bits (unless otherwisespecified).


Fig. 3. Comparison of different SIT-QCSIT and Q-CSIT schemes (N = 64,BER = 10 , R = 20, and B = 2).

Fig. 4. Comparison of D-CSIT and Q-CSIT schemes (N = 64, BER =

10 , R = 20, and B = 2).

Test Case 1 (SIT-QCSIT Schemes): Fig. 3 compares theoptimal and reduced complexity SIT-channel quantizers;namely, EP-QCSIT, Lloyd- , and Lloyd- schemes, forboth Tx-A and Tx-B adaptation strategies. As expected,Tx-A always outperforms the less complex Tx-B for allschemes. Less expected is the fact that all reduced com-plexity quantizers suffer minimal loss (at most 1 dB) inpower efficiency relative to the optimum one. In particular,EP-QCSIT performs the best among suboptimum quan-tizers (comes to within 0.5 dB close to the optimum); andfor this reason, we will henceforth use it as the representa-tive of this class.Test Case 2 (D-CSIT Versus Q-CSIT With StatisticallyDistinct or Equal Gains): Fig. 4 quantifies the powerloss when relying on Q-CSIT rather than D-CSIT. Threeschemes utilizing Q-CSIT are tested (optimum Q-CSIT,CT-QCSIT, and EP-QCSIT) for the different adaptivetransmitters TxA–D. We observe that all four strategiesTxA–D perform within 2 dB with respect to each other.

Fig. 5. Comparison of D-CSIT and Q-CSIT schemes with equal subchannelsstatistics (N = 64, BER = 10 , R = 20,B = 2, and �g = 2).

Fig. 6. Comparison of systems with different CSIT assumptions and BER con-straints (N = 64, R = 25, and B = 2).

In addition, EP-QCSIT shows near-optimum power ef-ficiency almost 1 dB away from the optimal D-CSITbenchmark with fully adaptive transmission (Tx-A). It isworth mentioning that the small variations in the BERcurve around the 10 target are due to finite simulationsamples.Fig. 5 is the counterpart of Fig. 4 for the same .As expected intuitively, in this case, the optimum Q-CSITscheme exhibits performance identical with CT-QCSIT,even though the two entail different sets of thresholds.This points out the possibility that the optimum solutionmay not be unique. Moreover, the equivalence of Tx-A andTx-B testifies the optimality of nearly uniform bit loadingacross subcarriers as the subchannels become statisticallyequivalent.Test Case 3 (D-CSIT Versus Q-CSIT Versus S-CSIT inPower Efficiency and BER): For variable BER specifi-cations, Fig. 6 includes also in the comparison optimalpower loading based on S-CSIT, as well as uniform power


Fig. 7. Comparison of D-CSIT and Q-CSIT systems with different BER con-straints (N = 64, R = 25, and B = 2).

(UP) loading based on S-CSIT. We observe that the (gen-erally nonuniformly loaded) optimum S-CSIT design isas efficient in terms of power as UP-SCSIT. Furthermore,compared with both S-CSIT-based designs, Q-CSIT ex-hibits an average power gain of approximately 8, 15, and24 dB for 10 , 10 , and 10 , respectively.Compared with the no-CSIT case, this gain almost dou-bles since the S-CSIT scheme has itself a power gain inthe range of 15–25 dB. A closer look offered by Fig. 7at both D-CSIT and Q-CSIT reveals that the optimumQ-CSIT scheme comes surprisingly close (about 1 dB) tothe D-CSIT scheme. In addition, the EP-QCSIT schemecomes within 1 dB close to the optimum Q-CSIT one. Itcan also be inferred that CT-QCSIT is inferior to all otherschemes especially for very small BER values.Test Case 4 (Reduced Complexity Optimization and Feed-back Overhead): Here, we test suboptimal schemes withcontrollable complexity and feedback overhead which relyon RC1-RC4 to simplify the optimization for loading andalso reduce the reverse-channel overhead by feeding backQ-CSI bits corresponding to the minimal number of sub-channel gains (recall that the remaining gains are obtainedvia interpolation). Fig. 8 depicts the power efficiency of thevarious options in comparison with D-CSIT and Q-CSITwithout reduced feedback overhead. Accuracy in the esti-mation and interpolation steps corresponding to differentRC alternatives manifests itself in the fluctuation of theBER around its nominal value. Thanks to its simplicity androbustness to estimation errors, the RC2 option with ampli-tude estimation is preferable when the region is given. No-tice also the negligible 0.3 dB loss in average total powerrelative to feedback-per-subchannel Q-CSIT (i.e., no feed-back reduction), and the minimal 1 dB loss compared withthe D-CSIT scheme.Test Case 5 (Number of Quantization Regions): We finallyinvestigate the effect of the number of feedback bits in theEP-QCSIT scheme. Fig. 9 demonstrates that power gains

Fig. 8. Sinc interpolator with different estimation methods (N = 64, R =20, B = 2, L = 3, [jh j ] = 1, [jh j ] = 0:5, [jh j ] = 0:25, and[jh j ] = 0:1).

Fig. 9. Effect of number of feedback bits using the EP-QCSIT scheme (N =64 and R = 25).

increase markedly when increases from 1 to 2 bits, whilea diminishing relative gain is expected for each additionalincrease in . It also shows that having entails only2 dB in power loss compared with the D-CSIT case whichcorresponds to .

IX. CONCLUDING SUMMARY AND FUTURE RESEARCH

Under prescribed rate and error probability constraints, weoptimized power-efficiency of wireless OFDM by relying onvarious forms of CSIT. Emphasis was placed on Q-CSIT whichcan become readily available because the receiver needs to feedback to the transmitter only a limited number of bits indexingthe quantization region which the underlying channel falls into.The resultant optimized transceivers hold great potential sincethey were shown to enjoy about 15 dB power savings relative toS-CSIT (and twice as much when compared with OFDM withno-CSIT available), while staying within 1 dB from D-CSITbenchmark designs that are more suitable for slow fading wire-less or wireline channels.


To choose a judicious form of Q-CSIT, we designed offlineoptimal channel quantizers that rely on channel statistics (mean)to select quantization thresholds across OFDM subchannels.We further derived reduced complexity quantizers which eitheradopt CT for all subchannels, or, select thresholds based on in-dividual subchannel statistics. Among the latter, the simplestone exhibiting near-optimal power savings turned out to be theone for which thresholds are selected per subchannel to renderquantization regions EP. OFDM based on EP-QCSIT was seento outperform those relying on scalar quantization based onLloyd’s algorithm, as well as those relying on CT-QCSIT.

With the designed Q-CSIT fixed, we also developed optimaland simpler suboptimal algorithms for online power and bitloading. Applying these to OFDM transmitters with variabledegrees of adaptation (e.g., subcarrier or bit adaptation only)offers flexibility to tradeoff power savings for ease in imple-mentation and reduced complexity online operation. To furthersimplify the operation of Q-CSIT-based designs, we relied oninterpolation to devise reduced overhead feedback schemeswhich require Q-CSIT only on a few subchannels (as manyas twice the length of the discrete-time equivalent channel).Among other comparisons, our simulations tested how thenumber of feedback bits affect performance and revealed thatsimple (EP) quantizers with 1 bit fed back per subchannel andtruncated sinc interpolation can afford no more than 2 dB lossin power relative to the D-CSIT benchmark.

Interesting directions not explored here include the use ofvector quantization for obtaining improved Q-CSIT at the priceof increased complexity, as well as the possibility of feedingback a possibly different number of bits per subchannel. Thepromise power-efficient OFDM holds in the single-antennapoint-to-point scenario considered here, motivates futureresearch to explore its potential for power savings whenmulticarrier modulation is used over multiuser and (possiblydistributed) multiantenna links in the context of ad hoc andwireless sensor networks.

APPENDIX AON THE OPTIMALITY OF THE QUANTIZER

While the KKT conditions are necessary, sufficientconditions for global optimality include the convexityof both objective and constraint functions [10]. Threeconditions have to be satisfied to ensure convexity of

, which is a

function of both and [10, Appendix 1]:, , and

. It is clear that .It is also straightforward to show that the third condition cannotbe satisfied; and hence, being nonconvex, the offlinesolution is only guaranteed to be locally optimal.

APPENDIX BPROOF OF PROPOSITION 1

The feedback information is a quantized version of the realparameter . With denoting the DFT of length and

representing a sequence of length such that, can be expressed as

(26)

where is the conjugate of , denotes the -circular con-volution, and stands for modulo . The correlationsequence can be shown to have the following structure(given that ; otherwise, it is trivial that we revert tothe feedback-per-subchannel case)

otherwise.(27)

Using this structure of and (26), any gain can beexpressed as

(28)

where and denote, respectively, the real and imagi-nary parts of . Therefore, since we have unknowns thesame number of coefficients suffices to fully describe .This result implies that all gains can be acquired if onlyof them are known. As a result, bits suffice to be usedfor feedback.

REFERENCES

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[7] A. Gersho and R. M. Gray, Vector Quantization and Signal Compres-sion. Norwell, MA: Kluwer, 1992.

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[15] T. K. Moon and W. C. Stirling, Mathematical Methods and Algorithmsfor Signal Processing. Upper Saddle River, NJ: Prentice-Hall, 2000.

[16] R. Nilsson, O. Edfors, M. Sandell, and P. O. Boerjesson, “An analysisof two-dimensional pilot-symbol assisted modulation for OFDM,” inProc. Int. Conf. Pers. Wireless Commun., Bombay, India, Dec. 1997,pp. 71–74.

[17] L. Piazzo, “Fast algorithm for power and bit allocation in OFDM sys-tems,” IEE Elect. Lett., vol. 35, pp. 2173–2174, Dec. 1999.

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Antonio G. Marqués (S’03) received the Degree inTelecommunication Engineering (B.Sc. and M.Sc.degrees in electrical engineering) with Honors fromthe Universidad Carlos III de Madrid, Madrid, Spain,in 2002. He is currently working towards the Ph.D.degree at the Universidad Carlos III de Madrid.

In 2003, he joined the Department of SignalTheory and Communications, Universidad Rey JuanCarlos, Madrid, Spain, where he currently developshis research and teaching activities. Since 2005, hehas been a Visiting Researcher at the Department of

Electrical Engineering, University of Minnesota, MN. His research interests liein the areas of communication theory, signal processing, and networking. Hiscurrent research focuses on time-frequency selective channels, channel stateinformation designs, energy-efficient resource allocation, and wireless ad hocand sensor networks.

Fadel F. Digham (S’95–M’06) was born in Egypt.He received the B.Sc. degree (Hon.) and the M.Sc.degree in electronics and communications from CairoUniversity, Cairo, Egypt, in 1995 and 1999, respec-tively, and the M.E.E. and Ph.D. degrees in electricalengineering from the University of Minnesota, Min-neapolis, in 2002 and 2005, respectively.

He worked for Alcatel-Telecom, Egypt branchfrom 1996 to 2000. He is currently a PostdoctoralResearcher with the Department of Electrical andComputer Engineering, University of Minnesota,

where his current research interests are tailored to wireless communicationsincluding resource allocation, wireless sensor networks, as well as distributedsignal processing and detection and estimation.

Georgios B. Giannakis (F’97) received the Diplomadegree in electrical engineering from the NationalTechnical University of Athens, Athens, Greece,in 1981. He received the M.Sc. degree in electricalengineering, the M.Sc. degree in mathematics, andthe Ph.D. degree in electrical engineering, fromthe University of Southern California (USC), LosAngeles, CA, in 1983, 1986, and 1986, respectively.

After lecturing for one year at USC, he joined theUniversity of Virginia in 1987, where he became aProfessor of Electrical Engineering in 1997. Since

1999, he has been a Professor with the Department of Electrical and Com-puter Engineering, University of Minnesota, Minneapolis, where he now holdsan ADC Chair in Wireless Telecommunications. His general interests span theareas of communications and signal processing, estimation and detection theory,time-series analysis, and system identification—subjects on which he has pub-lished more than 250 journal papers, 450 conference papers and four books. Cur-rent research focuses on diversity techniques for fading channels, complex-fieldand space-time coding, multicarrier, ultra-wide band wireless communicationsystems, cross-layer designs, and sensor networks.

Dr. Giannakis is the (co) recipient of six paper awards from the IEEE SignalProcessing (SP) and Communications Societies (1992, 1998, 2000, 2001, 2003,and 2004). He also received Technical Achievement Awards from the SP Societyin 2000 and from EURASIP in 2005. He served as Editor in Chief for the IEEESignal Processing Letters, as Associate Editor for the IEEE TRANSACTIONS ON

SIGNAL PROCESSING and the IEEE Signal Processing Letters, as Secretary of theSP Conference Board, as member of the SP Publications Board, as member andVice-Chair of the Statistical Signal and Array Processing Technical Committee,as Chair of the SP for Communications Technical Committee, and as a memberof the IEEE Fellows Election Committee. He has also served as a member ofthe IEEE-SP Society’s Board of Governors, the Editorial Board for the Pro-ceedings of the IEEE, and the steering committee of the IEEE TRANSACTIONS

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